Population of a certain species (in hundreds) evolves according to the autonomous…
MATHEMATICS · Multi-part · AP · Calculus BC · Differential Equations · Differential Equations
Problem
Population P(t) of a certain species (in hundreds) evolves according to the autonomous differential equation
dtdP=P(4−P),t measured in months.
An implicit relation for solutions of this differential equation is given by
ln∣P∣−ln∣4−P∣=4t+C,
where C is an arbitrary constant.
(a) Show that the implicit relation provided indeed generates solutions of the differential equation by differentiating implicitly and confirming the equality.
(b) Solve explicitly for P in terms of t and C. Simplify your expression as much as algebraically feasible.
(c) Given the initial condition P(0)=1, determine the particular solution and state its limiting value as t→∞. Provide justification.
(d) For the particular solution from part (c), determine the time at which the population is increasing most rapidly and compute that maximum growth rate. Explain your reasoning.